Generalized State Space Model (GSSM)

• Generalized State Space Model (GSSM) is a framework that uses convex sections and CP extensions to represent quantum channels and higher-order maps.
• It leverages Choi matrix characterizations to establish channel equivalence and enables precise structural analysis in quantum information theory.
• The iterative design of GSSM allows decomposition of complex quantum processes into simple, standard channel components for optimization.

A generalized state space model (GSSM) is a flexible mathematical and operator-theoretic framework for representing, classifying, and analyzing not only quantum states but also channels, instruments, and higher-order maps as convex sections of state spaces over finite-dimensional $C^*$-algebras. The concept encompasses “generalized channels,” their completely positive extensions, equivalence classes, and the iterative definition of higher-order maps (“supermaps”), forming a rigorously extensible hierarchy suitable for optimization and structural analysis in quantum information science.

1. Convex Sections and the Definition of Generalized Channels

Let $A$ and $B$ be finite-dimensional $C^*$-algebras, and let $G(A)$ denote the state space of $A$, that is, the set of positive trace-one linear functionals (states) on $A$. The starting point for GSSM is the selection of a convex subset $K \subset G(A)$, which serves as the base domain for “generalized” constructs.

A channel on $K$ is defined as an affine map $E : K \rightarrow G(B)$, i.e., $$E(\lambda \rho_1 + (1-\lambda) \rho_2) = \lambda E(\rho_1) + (1-\lambda) E(\rho_2)$$, that can be (uniquely) extended to a completely positive (cp) map on the vector subspace $[K]$ generated by $K$. The extension property is essential: the affine map on a convex “section” $K = [K]\cap G(A)$ uniquely determines, and is determined by, its cp extension to $[K]$ and, by further extension, to the full algebra $A$. This forms the basis for the “generalized channel”: an affine, cp-extendable map defined only on a partial convex section of a state space, not necessarily on all states.

Such generalized channels will, for every positive affine function on $K$, admit an extension to a positive linear map on $A$. This property is crucial for viewing $K$ as a section of a state space and for building further GSSM structure.

2. Choi Matrix Characterization and Equivalence of Channels

The characterization of generalized channels relies on their extension as cp maps and exploits the Choi–Jamiołkowski isomorphism. For every cp map $Q: A \rightarrow B$ representing a channel on $K$, its Choi matrix $X_Q$ must satisfy

$\operatorname{Tr}_B X_Q = I_A + (K^\top)^\perp,$

where $(K^\top)^\perp$ is the orthogonal complement (in the dual space) with respect to $K$. Equivalence of two such cp extensions $Q_1, Q_2$—meaning they have identical effects on $K$—is characterized by the Choi matrices through

$X_{Q_1} - X_{Q_2} \in B \otimes (K^\top)^\perp,$

that is, differences in the Choi representations live entirely in the tensor extension of the orthogonal complement to $K$. This explicit structure enables a classification of generalized channels and their equivalence classes, which is critical for both theoretical analysis and practical applications such as discrimination and optimization over channels.

3. The Role of Tracial States and the Iterative Structure of Supermaps

A key insight is that if the subset $K$ contains the tracial state $T_A$, the set of generalized channels, represented via their Choi matrices, itself forms a convex section of a higher-order multipartite state space (specifically, a section of $B \otimes A$). This inclusion promotes a “self-similar” or recursive character: the space of generalized channels becomes a new section $L$, and cp maps on $L$ (which can themselves be extended to cp maps on the larger algebra) define generalized supermaps, i.e., maps taking channels to channels in a convex-operator-theoretic sense.

This hierarchy can be iterated: supermaps on supermaps, and so forth. In the language of quantum information, such higher-order maps correspond to objects like quantum combs (multi-step quantum processes), process POVMs (tests on quantum processes), and instruments (generalized measurements on quantum channels).

4. Decomposition Theorem: Factorization into Simple and Standard Channels

A principal technical result is a decomposition theorem for generalized supermaps and channels. Specifically, every generalized channel $Q: A \rightarrow B$ (with respect to $K$) admits a canonical decomposition

$Q = A \circ X_c,$

where $X_c$ is a simple generalized channel parameterized by a positive element $c \in A_+$,

$X_c(a) = c^{1/2} a c^{1/2}$

subject to the normalization $$\operatorname{Tr} (X_c(a)) = \operatorname{Tr}(a c) = 1$$ for all $a \in K$, and $A: A \rightarrow B$ is an ordinary channel. This factorization separates the structural adjustment due to the restriction to $K$ (the $X_c$ component) from the “standard” part of the channel (the $A$ component). The extension to supermaps proceeds similarly, capturing the iterative structure of generalized state space models: complex higher-order maps can be expressed in terms of layered compositions of such simple and standard components.

The decomposition theorem provides a concrete, constructive method to analyze and build up arbitrary (possibly very complex) processes in a GSSM, reducing the classification and manipulation of such objects to well-understood pieces.

5. Special Cases: Quantum Combs, Process POVMs, and Instruments

The framework encompasses numerous familiar quantum information-theoretic objects as special cases:

Object Type	Subset $K$ or Target Algebra	GSSM Interpretation
Quantum combs	$K$ with tracial state	Generalized supermaps for multi-step strategies
Process POVMs	Commutative $C^*$ algebra $B$	Measurements/tests on processes via generalized POVMs
Quantum testers	$K$ as POVMs	Recovering structure of process measurements
Instruments	cp maps on commutative subalgebras	Measurements on generalized POVMs

When $B$ is commutative or $K$ is associated with a POVM structure, process POVMs and instruments arise as generalized POVMs in the GSSM sense, with Choi conditions and normalization constraints specialized accordingly.

6. Mathematical and Physical Implications in GSSM

The generalized state space model framework systematically unifies the representation of all basic objects in quantum information—in particular, states, channels, and higher-level processes—within the geometry of convex sections of operator algebras. Every affine map on such a section, if it extends to a cp map, is included, and even partial channels (those defined only on a subset $K$) can be analyzed via their cp extensions. This is particularly salient in optimization problems, process tomography, or resource analysis, where the convexity and operator structure are central.

The iterative, section-based approach enables hierarchical modeling: if $K$ contains the tracial state, the set of generalized channels is itself a section of a higher-dimensional state space, supporting the construction and decomposition of supermaps. This mirrors the hierarchies in quantum strategies and testing and allows powerful reduction theorems (as given in the decomposition result) to be used to break down any generalized process into constituent channels. These techniques extend the formalism to encompass quantum combs, testers, instruments, and more, articulating a fully unified mathematical language for generalized state space analysis.

7. Summary Table: Choi Constraints and Equivalence

Concept	Choi Condition / Equivalence Class
Generalized channel $Q$	$\operatorname{Tr}_B X_Q = I_A + (K^\top)^\perp$
Channel equivalence	$X_{Q_1} - X_{Q_2} \in B \otimes (K^\top)^\perp$
Supermap decomposition	$Q = A \circ X_c$, $X_c(a) = c^{1/2} a c^{1/2}$

This explicit algebraic characterization facilitates the practical manipulation, extension, and optimization of all levels of quantum processes within the generalized state space framework.

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In conclusion, the GSSM framework grounded in operator convexity and cp-extendability establishes a mathematically rigorous foundation for representing, classifying, and composing a broad hierarchy of quantum information processes, from states to channels to multi-level supermaps. It provides general decomposition theorems, explicit algebraic characterizations, and encompasses numerous known special cases, making it central to structural and optimization questions in quantum information and operator theory (Jencova, 2011).